Ga­tor­ade hairdo blow-dried into feathery thistles, Hayley Williams twirls around the stage like some punk rock bird of paradise, punching ghosts, spitting flame. Her band mates roar along, wearing denim mottled with rainbow splotches, like the Clash after a day of paintball.

But Paramore is emerging from a much messier war. The platinum-selling group first wrote its name in the sky in the mid-aughts with rock songs that felt brash and spontaneous, bursting into existence like arguments.

Some were actual arguments. Williams memorialized her band’s squabbles in more than a few lung-emptying refrains, giving rapacious young fans a front row seat to the band’s spiraling soap opera.

In 2010, it finally became too much for Josh and Zac Farro, two brothers who founded the band. So they quit. And then they blogged about it, entering an ugly crossfire of keystrokes with Williams, guitarist Taylor York and bassist Jeremy Davis.

Watching the breach play out on a computer screen was both sad and strange. This was a group that could turn its inner dramas into singalongs with the piquancy of Fleetwood Mac and the efficiency of Black Flag. But Paramore’s remaining members say they never considered shutting the whole thing down. The band’s vivacious new self-titled album, out Tuesday, finds the trio dusting each other off, talking a little trash and strutting into a brilliant, technicolor sunset.

“Even if I tried, I could never not have this coursing through my veins,” says Williams of the band she joined at 14.

It’s six hours before showtime in Austin — a midnight showcase at the South By Southwest music festival designed to reintroduce Paramore to the industry, the media, the fans, the universe. Backstage, the three huddle on a tufted sofa. Williams, 24, wears her orange hair in a Dutch crown braid, looking more like a sprite than the punk who joined Paramore in Franklin, Tenn., a decade ago.

“We all started a band as kids because we really just wanted to be in a band,” Williams says. “And [the Farros] ended up not wanting it as much as they thought they did. For me, it’s that simple. They weren’t happy. And you can’t fault someone for not being happy or for finding complete joy in the same things that you do.”

Williams has used this high-road party line in just about every interview since the split. She doesn’t mention the blog post where Josh Farro — whom Williams was dating when the band’s 2007 album, “Riot,” was on its way to platinum certification — accused his ex-girlfriend of compromising the band’s quiet Christian beliefs and letting her family meddle in Paramore’s dynamic.

“In reality, what started as natural somehow morphed into a manufactured product of a major label, riding on the coattails of ‘Hayley’s dream,’” Farro blogged. The angst was underscored by the fact that Williams was the only one in Paramore technically signed to Atlantic Records.

This all came a year after Williams threw her pipes into Atlanta rapper B.o.B.’s otherwise dishwater radio hit “Airplanes.” As the tune floated up the Billboard singles chart, fans wondered whether the singer was branding herself as a solo artist.

Williams, Davis and York insist that idea was never in play, and three years later, they have the best Paramore album ever recorded to prove it. With guidance from producer Justin Meldal-Johnsen, former bassist for Nine Inch Nails and Beck, the band has gone chasing after new textures and new tempos with no fear.

“I was constantly surprised by all the things we were getting away with,” Williams says of the band’s slog in the studio. “I don’t know if we’re aware of our boundaries at this moment. If something feels genuine to us, we’ll do it.”

As a three-piece band — live, they play with an auxiliary guitarist, a keyboardist and a drummer — there’s been less creative congestion and a galvanizing sense of freedom.

“We all want to be here. We all want to grow,” says York. “We have a better relationship with each other than ever and we’re more open to ideas, and trying things, and taking ourselves outside of the past.”

Over the new album’s 17 tracks, the trio negotiates a fresh reconciliation between rock crunch, emo bluster, R&B verve and pop gloss that should make Pink, Kelly Clarkson and Fall Out Boy all grind their molars.

Most satisfyingly, the latent R&B that’s always trickled through Paramore’s songbook has finally surfaced and bloomed. Williams’s voice has never sounded more powerful or elastic than it does today. With “Ain’t It Fun,” a new jack swing-flavored tune that York and Davis cite as a turning point in the band’s songwriting process, Williams holds her own alongside a gospel choir — something that should provoke eye rolls, but because it’s a Paramore song, magically doesn’t.

Elsewhere, the band starts to resemble the Cars and Tangerine Dream and Blondie andLiving Colour, but it never stops sounding like Paramore. That’s because loud, shameless, euphoric catharsis will always be its defining trait.

“If we’re gonna connect with people, we need to be honest,” says Williams, which means spilling your guts even if you accidentally “slip on ’em,” as she sings in one of her lyrical darts.

“It’s just like sitting down with a friend over coffee,” she says. “You talk about that thing you’re going through, the things you’re feeling, the things that are heaviest on your heart.”

The intoxicatingly mushy love songs on “Paramore” signal that life is good, but Williams hasn’t donated her poison pen collection to Goodwill. The album’s first three cuts address the band’s fissure head-on, culminating with “Grow Up,” perhaps the album’s most radio-ready kiss-off: “Some of us have to grow up sometimes . . . So if I have to, I’m gonna leave you behind.”

On stages and behind microphones, Williams projects fearlessness and ambition ungoverned. She might be the most ebullient rock-and-roll frontperson of her generation. But flanked by York and Davis backstage on the couch, she still seems the tiniest bit squirmy about Paramore’s first steps back into the pop wilderness.

“Everything is falling into place the way it’s supposed to,” she says. “And I have to believe that, or else the last two years would have been the most insane roller coaster for nothing. . . . All I can do is be excited right now. We got a second chance to be Paramore.”

She keeps her hands tucked in the sleeves of her polka-dot blouse, but her fists are clenched.

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